relerr_sf: Relative error scoring function (MAE-PROP scoring function)

In scoringfunctions: Loss Functions for Assessing Point Forecasts

Relative error scoring function (MAE-PROP scoring function)

Description

The function relerr_sf computes the relative error scoring function when y materialises and x is the predictive \textnormal{med}^{(1)}(F) functional.

The relative error scoring function is defined in Table 1 in Gneiting (2011).

The relative error scoring function is referred to as MAE-PROP scoring function in eq. (13) in Patton (2011).

Usage

relerr_sf(x, y)

Arguments

x	Predictive \textnormal{med}^{(1)}(F) functional (prediction). It can be a vector of length n (must have the same length as y).
y	Realisation (true value) of process. It can be a vector of length n (must have the same length as x).

Details

The relative error scoring function is defined by:

S(x, y) := |(x - y)/x|

Domain of function:

x > 0

y > 0

Range of function:

S(x, y) \geq 0, \forall x, y > 0

Value

Vector of relative errors.

Note

For details on the relative error scoring function, see Gneiting (2011).

The \beta-median functional, \textnormal{med}^{(\beta)}(F) is the median of a probability distribution whose density is proportional to y^\beta f(y), where f is the density of the probability distribution F of Y (Gneiting 2011).

The functional \textnormal{med}^{(1)}(F) elicited by the relative error scoring function is the \beta-median at \beta = 1, whose density is proportional to y f(y) (p. 748 and p. 749 in Gneiting 2011). The \beta-median functional at general \beta is elicited by bmedian_sf.

Patton (2011) adds that the MAE-PROP scoring function is also known as the mean absolute percentage error (MAPE). That name is used differently here: the package follows Gneiting (2011), where the absolute percentage error is |(x - y)/y|, elicited by aperr_sf and averaged by mape, whose elicited functional is \textnormal{med}^{(-1)}(F). A reader arriving from Patton (2011) should use relerr_sf, not mape.

The relative error scoring function is negatively oriented (i.e. the smaller, the better).

The relative error scoring function is strictly \mathbb{F}^{(w)}-consistent for the \textnormal{med}^{(1)}(F) functional. \mathbb{F} is the family of probability distributions for which \textnormal{E}_F[Y] exists and is finite. \mathbb{F}^{(w)} is the subclass of probability distributions in \mathbb{F}, which are such that w(y) f(y), w(y) = y has finite integral over (0, \infty), and the probability distribution F^{(w)} with density proportional to w(y) f(y) belongs to \mathbb{F} (see Theorems 5 and 9 in Gneiting 2011).

References

Gneiting T (2011) Making and evaluating point forecasts. Journal of the American Statistical Association 106(494):746–762. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/jasa.2011.r10138")}.

Patton AJ (2011) Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160(1):246–256. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jeconom.2010.03.034")}.

See Also

mre

Examples

# Compute the relative error scoring function.

df <- data.frame(
    y = rep(x = 2, times = 3),
    x = 1:3
)

df$relative_error <- relerr_sf(x = df$x, y = df$y)

print(df)

scoringfunctions documentation built on Aug. 30, 2026, 5:07 p.m.